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A337686 a(n) is the least multiplier k such that n*k has twice as many divisors as n.

%I A337686 #40 Jun 20 2025 08:12:15
%S A337686 2,3,2,3,2,4,2,3,2,3,2,5,2,3,2,3,2,4,2,3,2,3,2,5,2,3,2,3,2,4,2,3,2,3,
%T A337686 2,5,2,3,2,3,2,4,2,3,2,3,2,5,2,3,2,3,2,4,2,3,2,3,2,6,2,3,2,3,2,4,2,3,
%U A337686 2,3,2,5,2,3,2,3,2,4,2,3,2,3,2,5,2,3,2,3,2,4
%N A337686 a(n) is the least multiplier k such that n*k has twice as many divisors as n.
%C A337686 The zeros in A139315 are the missing values in this sequence (see A337709).
%C A337686 There are no 1's in this sequence. a(n) = 2 for all odd n and a(n) >= 3 for all even n. - _J. Lowell_, Sep 15 2020
%C A337686 Empirical observation: A007978(n) - a(n) = 1 for n = 60*A206547(n), = 2 for n = 420*A007310(n), else = 0. - _Hugo Pfoertner_, Sep 30 2020
%H A337686 Michel Marcus, <a href="/A337686/b337686.txt">Table of n, a(n) for n = 1..10000</a>
%H A337686 Hugo Pfoertner, <a href="/plot2a?name1=A007978&amp;name2=A337686&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">Illustration of ratio A007978(n) / a(n)</a>, using Plot 2.
%F A337686 a(n) = A129902(n)/n.
%e A337686 a(1) = 2 because 1 has 1 divisor, 1*2 has 2 divisors, so 2 is the least multiplier to apply to 1 to get twice as many divisors.
%t A337686 nn = 105; Do[d[i] = DivisorSigma[0, i], {i, 12 nn}]; Reap[Do[m = 2; While[d[m i] != 2 d[i], m++]; Sow[m ], {i, nn}]][[-1, -1]] (* _Michael De Vlieger_, Jan 10 2022 *)
%o A337686 (PARI) a(n) = {my(k=1); while (numdiv(n*k) != 2*numdiv(n), k++); k;}
%Y A337686 Cf. A000005, A129902, A139315, A337709 (missing values).
%K A337686 nonn
%O A337686 1,1
%A A337686 _Michel Marcus_, Sep 15 2020